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# 数学代写|实分析代写Real Analysis代考|MATH351 L’Hospital’s Rule

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## 数学代写|实分析代写Real Analysis代考|L’Hospital’s Rule

It is a familiar story that the Marquis de L’Hospital (1661-1704) stole what has come to be known as L’Hospital’s rule from Johann Bernoulli. It needs to be tempered with the observation that while the result is almost certainly Bernoulli’s, L’Hospital was a respectable mathematician who had paid for the privilege of publishing Bernoulli’s results under his own name.
To learn more about the Marquis de l’Hospital, his role in the early development of calculus, our uncertainty over how to spell his name, and why we do not pronounce the ” $s$ ” in his name, go to The Marquis de l’Hospital.
We work with the Archimedean definition of limit given on page 59. We also need to be careful about what we mean by an infinite limit and a limit at infinity.

Definition: infinite limit and limit at infinity
The statement
$$\lim {x \rightarrow a} f(x)=\infty$$ means that for any real number $L$, we can force $f(x)>L$ by restricting $x$ to be sufficiently close to $a$. That is to say, there is a $\delta>0$ so that $|x-a|<\delta$ implies that $f(x)>L$. When we write $$\lim {x \rightarrow \infty} f(x)=T$$
we mean that for any positive $\epsilon$, we can force $f(x)$ to be within $\epsilon$ of $T$ by taking $x$ sufficiently large. In other words, there is an $N$ so that $x>N$ implies that $|f(x)-T|<$ $\epsilon$

## 数学代写|实分析代写Real Analysis代考|Intermediate Value Property for Derivatives

In exercise 3.1.14 of section 3.1, you were asked to prove that if $\lim _{x \rightarrow a} f^{\prime}(x)$ exists, then so does $f^{\prime}(a)$, and they must be equal. This implies that where the limit exists, the derivative must be continuous. Gaston Darboux (1842-1917) was the first to observe that even more is true. Even if a derivative is not continuous, it must have the intermediate value property. By Theorem 3.4, the modified converse of the intermediate value theorem, if a derivative is not continuous then it cannot be piecewise monotonic. All examples of discontinuous derivatives are similar to the derivative of $x^2 \sin \left(x^{-1}\right)$ which exists but is not continuous at $x=0$ because the derivative oscillates infinitely often in any neighborhood of 0 . Our proof is based on one discovered by Lars Olsen.

Theorem 3.14 (Darboux’s Theorem). If $f$ is differentiable on $[a, b]$, then $f^{\prime}$ has the intermediate value property on $[a, b]$.
Proof: We define a new function $g$ :
$$g(x)= \begin{cases}f^{\prime}(a), & x=a, \ \frac{f(2 x-a)-f(a)}{2 x-2 a}, & a<x \leq(a+b) / 2, \ \frac{f(b)-f(2 x-b)}{2 b-2 x}, & (a+b) / 2 \leq x<b, \ f^{\prime}(b), & x=b .\end{cases}$$

.

## 数学代写|实分析代写Real Analysis代考|L’Hospital’s Rule

$$\lim x \rightarrow a f(x)=\infty$$

$$\lim x \rightarrow \infty f(x)=T$$

## 数学代写|实分析代写Real Analysis代考|Intermediate Value Property for Derivatives

$$g(x)=\left{f^{\prime}(a), \quad x=a, \frac{f(2 x-a)-f(a)}{2 x-2 a}, \quad a<x \leq(a+b) / 2, \frac{f(b)-f(2 x-b)}{2 b-2 x}, \quad(a+b) / 2 \leq x<b, f^{\prime}(b), \quad x=b .\right.$$

## MATLAB代写

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